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Step-by-Step Solution
Step 1: Rewrite the differential equation
The given differential equation is:
$e^x \sqrt{1 - y^2}\,dx + \frac{y}{x}\,dy = 0.$
Rearrange it to isolate terms involving $dx$ on one side:
$e^x \sqrt{1 - y^2}\,dx = -\,\frac{y}{x}\,dy.$
Step 2: Propose a form for the solution
Observe that the derivative of $\sqrt{1 - y^2}$ with respect to $y$ is
$-\frac{y}{\sqrt{1 - y^2}}.$
This suggests that $\sqrt{1 - y^2}$ might appear as part of a simpler expression when we differentiate with respect to $x$.
A natural guess, guided by the structure, is that
$\sqrt{1 - y^2} = e^x \,(x - 1) + C,$
where $C$ is a constant of integration.
Step 3: Apply the initial condition to find the constant
We are given $y(1) = -1.$ Substituting $x = 1$ and $y = -1$ into
$\sqrt{1 - y^2} = e^x\,(x - 1) + C$:
$\sqrt{1 - (-1)^2} = \sqrt{1 - 1} = 0 \quad\longrightarrow\quad
e^1 \,(1 - 1) + C = 0 + C = 0.$
Hence, $C = 0.$
Thus, the solution simplifies to
$\sqrt{1 - y^2} = e^x\,(x - 1).$
Step 4: Find $y^2$ at $x = 3$
Substitute $x = 3$ into the solution:
$\sqrt{1 - y^2} = e^3 \,(3 - 1) = 2\,e^3.
Square both sides to solve for $y^2$:
$1 - y^2 = \bigl(2\,e^3\bigr)^2 = 4\,e^6
\quad\Longrightarrow\quad
y^2 = 1 - 4\,e^6.
Step 5: Conclude the value of $(y(3))^2$
The value of $(y(3))^2$ is therefore
$\boxed{1 - 4\,e^6}.
